solving investment problem

Question

I need the answer to this question and how to work it out. 

 

An artist sold a painting for $410,000. He needs some of the money in 6 months and the rest in 1 yr. He can get a treasury bond for 6 months at 2.62% and one for a year at 2.91%. His broker tells him the two investments will earn at total of $8761. How much should be invested at each rate to obtain that amount of interest? 

Answer 1

ASSUMING that this is just a simple interest problem and NOT a compounded interest problem:

 

Let: P₁ = Money invested for 6 months; and P₂ = Money invested for 1 year.

 

From this, the first equation is simply the sum of the two which is $410,000:

 

P₁ + P₂ = 410000, eq. 1

 

Next: Let I₁ = Interest from P₁; and I₂ = Interest from P₂

 

The same as the previous one, the sum of the interests is $8761, therefore:

 

I₁ + I₂ = 8761, eq. 2

 

Now, we have two equations but with different sets of variables. However we do know that:

 

I = Prt, Meaning:

 

I₁ = P₁r₁t₁ = P₁(.0262)(0.5)      //the 0.5 is the 6 months in terms of years//

I₁ = .0131P₁

 

Doing the same with I₂, with r₂ = 0.0291 and t₁ = 1,

 

I₂ = .0291P₂

 

Substituting I₁ and I₂ to eq. 2, you'll get:

 

.0131P₁ + .0291P₂ = 8761      //you can multiply both sides by 100 to make it a little prettier//

 

1.31P₁ + 2.91P₂ = 876100, eq. 3

  

P₁ + P₂ = 410000, eq. 1    //using eq. 1 and eq. 3, through either substitution or elimination//

 

P₁ = $198125

P₂ = $211875